Second order linear ODE: Difference between revisions
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(Created page with "Second order linear ODEs are in the following form: <math> y''+p(t)y'+q(t)y=g(t) </math> Important types of second order linear ODEs include * Homogeneous * Constant coefficients (where p and q are constants) = Initial value problem = There are two arbitrary constants in the solution of a second order linear ODE, so we need two initial conditions. <math> \begin{cases} y(t_0)=y_0\\ y'(t_0)=y_0' \end{cases} </math> = Solutions = == Constant coefficient, homogeneou...") |
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<math> | <math> | ||
y=e^{rt} | y=e^{rt} | ||
</math> | |||
We substitute in the guess and obtain the characteristic equation | |||
<math> | |||
\begin{aligned} | |||
ar^2e^{rt}+bre^{rt}+ce^{rt}&=0\\ | |||
ar^2+br+c&=0 | |||
\end{aligned} | |||
</math> | </math> | ||
[[Category:Differential Equations]] | [[Category:Differential Equations]] |
Revision as of 21:31, 1 May 2024
Second order linear ODEs are in the following form:
Important types of second order linear ODEs include
- Homogeneous
- Constant coefficients (where p and q are constants)
Initial value problem
There are two arbitrary constants in the solution of a second order linear ODE, so we need two initial conditions.
Solutions
Constant coefficient, homogeneous
These are the simplest kind. They have the general form
An exponential function has the property of being the same after many differentiation. We take advantage of this property and guess the solution to be the form of
We substitute in the guess and obtain the characteristic equation